Page 1: L'Hôpital's Rule

Introduction to L'Hôpital's Rule

• L'Hôpital's Rule:

  • Used for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞.

  • If the limit of f(x) as x approaches C produces g(x) in any indeterminate form, then:

  • [ \lim_{x \to C} \frac{f(x)}{g(x)} = \lim_{x \to C} \frac{f'(x)}{g'(x)} ] provided this limit exists or is infinite.

Example Evaluation

Example: ( \lim_{x \to 0^+} \frac{e^{-1/x}}{2x} )

• By direct substitution:

  • ( f(0) = 0, g(0) = 0 ), results in indeterminate form 0/0.

  • Apply L'Hôpital's Rule:

  • Derivatives:

    • ( f'(x) = e^{-1/x} \cdot \frac{1}{x^2} )

    • ( g'(x) = 2 )

• Rewrite limit:

  • [ \lim_{x \to 0^+} \frac{e^{-1/x}}{2x} = \lim_{x \to 0^+} \frac{e^{-1/x}}{2} ]

  • As x approaches 0, the limit approaches 0.

Page 2: Further Applications of L'Hôpital's Rule

Example Evaluation: ( \lim_{x \to \infty} \frac{\ln x}{x} )

• Indeterminate form: ( \frac{\infty}{\infty} )

• Apply L'Hôpital's Rule:

  • Derivative of the numerator: ( f'(x) = \frac{1}{x} )

  • Derivative of the denominator: ( g'(x) = 1 )

• Rewrite limit:

  • [ \lim_{x \to \infty} \frac{\ln x}{x} = \lim_{x \to \infty} \frac{1/x}{1} = 0 ]

Multiple Applications of L'Hôpital's Rule

Example: ( \lim_{x \to 0} \frac{x^2 e^{-x}}{1-x^2} )

• Indeterminate form: 0/0.

• Apply L'Hôpital's Rule multiple times:

  • Continue applying until a determinate form is reached.

Page 3: More Limit Evaluations

Example Evaluation

a) ( \lim_{x \to 3} \frac{e^x - 10}{x^2 - 9} )

• Indeterminate form: 0/0.

• Apply L'Hôpital's Rule:

  • Rewrite and evaluate derivatives to find limit.

b) ( \lim_{x \to 0^+} \frac{\ln x}{x} )

• Indeterminate form: ( -\infty/0 ) leads to limit evaluation approaching

  • Use derivatives to find the actual limit.

Page 4: Continued Evaluations of Limits

Key Evaluations

• Always check for possible patterns leading to 0 or ∞ when faced with indeterminate forms.

  • Remember to apply L'Hôpital’s multiple times if necessary.

Page 5: Special Notes on Indeterminate Forms

Note on Forms

• Indeterminate forms can often be rewritten to facilitate limit evaluation.

• Types of forms include: 0/0 and ∞/∞, which can often appear when substituting and finding limits.

Example Evaluation

Indeterminate Form ( 0 imes \infty )

Evaluate: ( \lim_{x \to 0^+} e^{-x} x )

• Requires rewriting into a fraction form: ( \lim_{x \to 0^+} \frac{x}{e^{x}} ) will help apply L'Hôpital's Rule.

Page 6: General Guidelines for Limit Evaluation

Transitioning Forms

• When faced with an indeterminate form:

  • Consider taking natural logarithms of both sides to simplify.

Example: ( \ln (1+x) )

• Evaluate limit:

  • Rewrite to generate a clearer path to solving the limit through logarithmic identities.

Page 7: Special Case Evaluations

Example with Sine

• Sine limits: When finding limits of forms involving ( sin(x) )

  • Focus on maximizing the utility of small angle approximations to establish limits approaching 0.

  • Example: ( \lim_{x \to 0} \frac{sin(x)}{x} = 1 )

Page 8: Final Evaluations and Conclusions

Example Summaries

• Example evaluations should incorporate reflections on initial forms and resultant derivatives.

• Summarize evaluations as applicable to typical calculus problems involving L'Hôpital's Rule.